Problem: Find the area of triangle $ABC$ below.


[asy]

unitsize(1inch);

pair P,Q,R;

P = (0,0);

Q= (sqrt(3),0);

R = (0,1);

draw (P--Q--R--P,linewidth(0.9));

draw(rightanglemark(Q,P,R,3));

label("$A$",P,S);

label("$B$",Q,S);

label("$C$",R,N);

label("$6$",R/2,W);

label("$30^\circ$",(1.25,0),N);

[/asy]
Since $ABC$ is a 30-60-90 triangle, we have $AB = AC\sqrt{3} = 6\sqrt{3}$, and  \[[ABC] = \frac{(AB)(AC)}{2} = \frac{(6)(6\sqrt{3})}{2} = \frac{36\sqrt{3}}{2} = \boxed{18\sqrt{3}}.\]